mirror of
https://github.com/ceres-solver/ceres-solver.git
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Sameer Agarwal
0141ca090c
Add [[deprecate]] notices to everything LocalParameterization related. Make sure that Ceres can be compiled without triggering deprecation warnings. Update the documentation: a. Add deprecation notices. b. Document interaction between LocalParameterization and Manifold coexisting in the Problem. c. Add documentation for Manifold(s) Change-Id: Ie4ad48963c83fded86e533c8c60561af402fbaff
205 lines
9.0 KiB
ReStructuredText
205 lines
9.0 KiB
ReStructuredText
.. highlight:: c++
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.. default-domain:: cpp
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.. _chapter-gradient_tutorial:
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==================================
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General Unconstrained Minimization
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==================================
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Ceres Solver besides being able to solve non-linear least squares
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problem can also solve general unconstrained problems using just their
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objective function value and gradients. In this chapter we will see
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how to do this.
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Rosenbrock's Function
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=====================
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Consider minimizing the famous `Rosenbrock's function
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<http://en.wikipedia.org/wiki/Rosenbrock_function>`_ [#f1]_.
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The simplest way to minimize is to define a templated functor to
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evaluate the objective value of this function and then use Ceres
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Solver's automatic differentiation to compute its derivatives.
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We begin by defining a templated functor and then using
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``AutoDiffFirstOrderFunction`` to construct an instance of the
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``FirstOrderFunction`` interface. This is the object that is
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responsible for computing the objective function value and the
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gradient (if required). This is the analog of the
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:class:`CostFunction` when defining non-linear least squares problems
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in Ceres.
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.. code::
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// f(x,y) = (1-x)^2 + 100(y - x^2)^2;
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struct Rosenbrock {
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template <typename T>
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bool operator()(const T* parameters, T* cost) const {
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const T x = parameters[0];
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const T y = parameters[1];
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cost[0] = (1.0 - x) * (1.0 - x) + 100.0 * (y - x * x) * (y - x * x);
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return true;
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}
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static ceres::FirstOrderFunction* Create() {
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constexpr int kNumParameters = 2;
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return new ceres::AutoDiffFirstOrderFunction<Rosenbrock, kNumParameters>(
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new Rosenbrock);
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}
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};
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Minimizing it then is a straightforward matter of constructing a
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:class:`GradientProblem` object and calling :func:`Solve` on it.
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.. code::
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double parameters[2] = {-1.2, 1.0};
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ceres::GradientProblem problem(Rosenbrock::Create());
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ceres::GradientProblemSolver::Options options;
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options.minimizer_progress_to_stdout = true;
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ceres::GradientProblemSolver::Summary summary;
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ceres::Solve(options, problem, parameters, &summary);
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std::cout << summary.FullReport() << "\n";
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Executing this code results, solve the problem using limited memory
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`BFGS
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<http://en.wikipedia.org/wiki/Broyden%E2%80%93Fletcher%E2%80%93Goldfarb%E2%80%93Shanno_algorithm>`_
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algorithm.
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.. code-block:: bash
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0: f: 2.420000e+01 d: 0.00e+00 g: 2.16e+02 h: 0.00e+00 s: 0.00e+00 e: 0 it: 2.29e-05 tt: 2.29e-05
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1: f: 4.280493e+00 d: 1.99e+01 g: 1.52e+01 h: 2.01e-01 s: 8.62e-04 e: 2 it: 8.39e-05 tt: 1.62e-04
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2: f: 3.571154e+00 d: 7.09e-01 g: 1.35e+01 h: 3.78e-01 s: 1.34e-01 e: 3 it: 2.22e-05 tt: 1.91e-04
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3: f: 3.440869e+00 d: 1.30e-01 g: 1.73e+01 h: 1.36e-01 s: 1.00e+00 e: 1 it: 5.01e-06 tt: 2.01e-04
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4: f: 3.213597e+00 d: 2.27e-01 g: 1.55e+01 h: 1.06e-01 s: 4.59e-01 e: 1 it: 3.81e-06 tt: 2.10e-04
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5: f: 2.839723e+00 d: 3.74e-01 g: 1.05e+01 h: 1.34e-01 s: 5.24e-01 e: 1 it: 4.05e-06 tt: 2.19e-04
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6: f: 2.448490e+00 d: 3.91e-01 g: 1.29e+01 h: 3.04e-01 s: 1.00e+00 e: 1 it: 5.01e-06 tt: 2.28e-04
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7: f: 1.943019e+00 d: 5.05e-01 g: 4.00e+00 h: 8.81e-02 s: 7.43e-01 e: 1 it: 4.05e-06 tt: 2.36e-04
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8: f: 1.731469e+00 d: 2.12e-01 g: 7.36e+00 h: 1.71e-01 s: 4.60e-01 e: 2 it: 1.22e-05 tt: 2.52e-04
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9: f: 1.503267e+00 d: 2.28e-01 g: 6.47e+00 h: 8.66e-02 s: 1.00e+00 e: 1 it: 5.96e-06 tt: 2.66e-04
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10: f: 1.228331e+00 d: 2.75e-01 g: 2.00e+00 h: 7.70e-02 s: 7.90e-01 e: 1 it: 4.05e-06 tt: 2.75e-04
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11: f: 1.016523e+00 d: 2.12e-01 g: 5.15e+00 h: 1.39e-01 s: 3.76e-01 e: 2 it: 9.06e-06 tt: 2.88e-04
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12: f: 9.145773e-01 d: 1.02e-01 g: 6.74e+00 h: 7.98e-02 s: 1.00e+00 e: 1 it: 5.01e-06 tt: 2.97e-04
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13: f: 7.508302e-01 d: 1.64e-01 g: 3.88e+00 h: 5.76e-02 s: 4.93e-01 e: 1 it: 5.01e-06 tt: 3.05e-04
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14: f: 5.832378e-01 d: 1.68e-01 g: 5.56e+00 h: 1.42e-01 s: 1.00e+00 e: 1 it: 4.77e-06 tt: 3.13e-04
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15: f: 3.969581e-01 d: 1.86e-01 g: 1.64e+00 h: 1.17e-01 s: 1.00e+00 e: 1 it: 4.05e-06 tt: 3.20e-04
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16: f: 3.171557e-01 d: 7.98e-02 g: 3.84e+00 h: 1.18e-01 s: 3.97e-01 e: 2 it: 8.82e-06 tt: 3.33e-04
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17: f: 2.641257e-01 d: 5.30e-02 g: 3.27e+00 h: 6.14e-02 s: 1.00e+00 e: 1 it: 4.05e-06 tt: 3.42e-04
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18: f: 1.909730e-01 d: 7.32e-02 g: 5.29e-01 h: 8.55e-02 s: 6.82e-01 e: 1 it: 1.00e-05 tt: 4.64e-04
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19: f: 1.472012e-01 d: 4.38e-02 g: 3.11e+00 h: 1.20e-01 s: 3.47e-01 e: 2 it: 1.29e-05 tt: 4.87e-04
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20: f: 1.093558e-01 d: 3.78e-02 g: 2.97e+00 h: 8.43e-02 s: 1.00e+00 e: 1 it: 5.01e-06 tt: 4.97e-04
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21: f: 6.710346e-02 d: 4.23e-02 g: 1.42e+00 h: 9.64e-02 s: 8.85e-01 e: 1 it: 4.05e-06 tt: 5.06e-04
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22: f: 3.993377e-02 d: 2.72e-02 g: 2.30e+00 h: 1.29e-01 s: 4.63e-01 e: 2 it: 1.00e-05 tt: 5.25e-04
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23: f: 2.911794e-02 d: 1.08e-02 g: 2.55e+00 h: 6.55e-02 s: 1.00e+00 e: 1 it: 5.01e-06 tt: 5.34e-04
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24: f: 1.457683e-02 d: 1.45e-02 g: 2.77e-01 h: 6.37e-02 s: 6.14e-01 e: 1 it: 4.05e-06 tt: 5.42e-04
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25: f: 8.577515e-03 d: 6.00e-03 g: 2.86e+00 h: 1.40e-01 s: 1.00e+00 e: 1 it: 3.81e-06 tt: 5.49e-04
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26: f: 3.486574e-03 d: 5.09e-03 g: 1.76e-01 h: 1.23e-02 s: 1.00e+00 e: 1 it: 4.05e-06 tt: 5.57e-04
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27: f: 1.257570e-03 d: 2.23e-03 g: 1.39e-01 h: 5.08e-02 s: 1.00e+00 e: 1 it: 3.81e-06 tt: 5.65e-04
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28: f: 2.783568e-04 d: 9.79e-04 g: 6.20e-01 h: 6.47e-02 s: 1.00e+00 e: 1 it: 4.05e-06 tt: 5.73e-04
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29: f: 2.533399e-05 d: 2.53e-04 g: 1.68e-02 h: 1.98e-03 s: 1.00e+00 e: 1 it: 4.05e-06 tt: 5.81e-04
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30: f: 7.591572e-07 d: 2.46e-05 g: 5.40e-03 h: 9.27e-03 s: 1.00e+00 e: 1 it: 5.96e-06 tt: 6.30e-04
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31: f: 1.902460e-09 d: 7.57e-07 g: 1.62e-03 h: 1.89e-03 s: 1.00e+00 e: 1 it: 4.05e-06 tt: 6.39e-04
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32: f: 1.003030e-12 d: 1.90e-09 g: 3.50e-05 h: 3.52e-05 s: 1.00e+00 e: 1 it: 3.81e-06 tt: 6.47e-04
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33: f: 4.835994e-17 d: 1.00e-12 g: 1.05e-07 h: 1.13e-06 s: 1.00e+00 e: 1 it: 4.05e-06 tt: 6.59e-04
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34: f: 1.885250e-22 d: 4.84e-17 g: 2.69e-10 h: 1.45e-08 s: 1.00e+00 e: 1 it: 4.05e-06 tt: 6.67e-04
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Solver Summary (v 2.0.0-eigen-(3.3.9)-lapack-suitesparse-(5.8.1)-cxsparse-(3.2.0)-acceleratesparse-eigensparse-no_openmp-no_custom_blas)
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Parameters 2
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Line search direction LBFGS (20)
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Line search type CUBIC WOLFE
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Cost:
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Initial 2.420000e+01
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Final 1.955192e-27
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Change 2.420000e+01
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Minimizer iterations 36
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Time (in seconds):
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Cost evaluation 0.000000 (0)
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Gradient & cost evaluation 0.000008 (44)
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Polynomial minimization 0.000067
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Total 0.000721
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Termination: CONVERGENCE (Parameter tolerance reached. Relative step_norm: 1.890726e-11 <= 1.000000e-08.)
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Initial x: -1.2 y: 1
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Final x: 1 y: 1
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If you are unable to use automatic differentiation for some reason
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(say beacause you need to call an external library), then you can
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use numeric differentiation. In that case the functor is defined as
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follows [#f2]_.
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.. code::
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// f(x,y) = (1-x)^2 + 100(y - x^2)^2;
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struct Rosenbrock {
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bool operator()(const double* parameters, double* cost) const {
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const double x = parameters[0];
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const double y = parameters[1];
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cost[0] = (1.0 - x) * (1.0 - x) + 100.0 * (y - x * x) * (y - x * x);
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return true;
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}
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static ceres::FirstOrderFunction* Create() {
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constexpr int kNumParameters = 2;
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return new ceres::NumericDiffFirstOrderFunction<Rosenbrock,
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ceres::CENTRAL,
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kNumParameters>(
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new Rosenbrock);
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}
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};
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And finally, if you would rather compute the derivatives by hand (say
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because the size of the parameter vector is too large to be
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automatically differentiated). Then you should define an instance of
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`FirstOrderFunction`, which is the analog of :class:`CostFunction` for
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non-linear least squares problems [#f3]_.
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.. code::
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// f(x,y) = (1-x)^2 + 100(y - x^2)^2;
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class Rosenbrock final : public ceres::FirstOrderFunction {
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public:
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~Rosenbrock() override {}
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bool Evaluate(const double* parameters,
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double* cost,
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double* gradient) const override {
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const double x = parameters[0];
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const double y = parameters[1];
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cost[0] = (1.0 - x) * (1.0 - x) + 100.0 * (y - x * x) * (y - x * x);
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if (gradient) {
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gradient[0] = -2.0 * (1.0 - x) - 200.0 * (y - x * x) * 2.0 * x;
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gradient[1] = 200.0 * (y - x * x);
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}
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return true;
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}
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int NumParameters() const override { return 2; }
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};
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.. rubric:: Footnotes
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.. [#f1] `examples/rosenbrock.cc
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<https://ceres-solver.googlesource.com/ceres-solver/+/master/examples/rosenbrock.cc>`_
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.. [#f2] `examples/rosenbrock_numeric_diff.cc
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<https://ceres-solver.googlesource.com/ceres-solver/+/master/examples/rosenbrock_numeric_diff.cc>`_
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.. [#f3] `examples/rosenbrock_analytic_diff.cc
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<https://ceres-solver.googlesource.com/ceres-solver/+/master/examples/rosenbrock_analytic_diff.cc>`_
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